• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,079 questions , 2,229 unanswered
5,348 answers , 22,758 comments
1,470 users with positive rep
819 active unimported users
More ...

  $\pm$ (light-cone?) notation in supersymmetry

+ 4 like - 0 dislike
  • I would like to know what is exactly meant when one writes $\theta^{\pm}, \bar{\theta}^\pm, Q_{\pm},\bar{Q}_{\pm},D_{\pm},\bar{D}_{\pm}$.

{..I typically encounter this notation in literature on $2+1$ dimensional SUSY like super-Chern-Simon's theory..}

  • I guess that when one has only half the super-space (i.e only the $+$ of the above or the $-$) it is called the $(0,2)$ superspace compared to the usual $(2,2)$ superspace. In this case of $(0,2)$ SUSY I have seen the following definitions,

$Q_+ = \frac{\partial}{\partial \theta^+} + i \bar{\theta}^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$

$\bar{Q}_+ = -\frac{\partial}{\partial \bar{\theta}^+} - i \theta^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$

which commute with,

$D_+ = \frac{\partial}{\partial \theta^+} - i \bar{\theta}^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$

$\bar{D}_+ = -\frac{\partial}{\partial \bar{\theta}^+} + i \theta^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$

  • I am guessing that there is an exactly corresponding partner to the above equations with $+$ replaced by $-$. Right?

How does the above formalism compare to the more familiar version as,

$Q_\alpha = \frac{\partial}{\partial \theta^\alpha} - i\sigma^\mu_{\alpha \dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\frac{\partial}{\partial x^\mu}$

$\bar{Q}_\dot{\alpha} = -\frac{\partial}{\partial \bar{\theta}^\dot{\alpha}} + i\sigma^\mu_{\alpha \dot{\alpha}}\theta^{\alpha}\frac{\partial}{\partial x^\mu}$

which commute with,

$D_\alpha = \frac{\partial}{\partial \theta^\alpha} + i\sigma^\mu_{\alpha \dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\frac{\partial}{\partial x^\mu}$

$\bar{D}_\dot{\alpha} = -\frac{\partial}{\partial \bar{\theta}^\dot{\alpha}} - i\sigma^\mu_{\alpha \dot{\alpha}}\theta^{\alpha}\frac{\partial}{\partial x^\mu}$

{..compared to the above conventional setting, in the $\pm$ notation among many things the most perplexing is the absence of the Pauli matrices!..why?..}

I would be very grateful if someone can explain this notation.

{..often it turns out that not just the Qs and the Ds but also various superfields also acquaire a $\pm$ subscript and various usual factors of Pauli matrices look missing..it would be great if someone can help clarify this..}

This post has been migrated from (A51.SE)
asked Mar 24, 2012 in Theoretical Physics by user6818 (960 points) [ no revision ]

1 Answer

+ 2 like - 0 dislike

It's hard to answer because you don't give many details, but since $\tfrac \partial {\partial y^2}$ does not appear in the SUSY generators, I think it's safe to assume that this is some light-cone superspace. In this kind of superspace one projects the spinors to plus and minus components with $\gamma_+ \gamma_-$ (I omitted a numerical factor which makes this a projector and which depends on your conventions).

Then the superspace is realized with only half of the odd variables, $\theta_+$ or $\theta_-$. So the only supercharges (or covariant derivatives) which are considered in this formalism are $\theta_+$ in your case. You don't need to include their analogs with $\theta_-$. If you insist on including them, you will generally run into problems with the closure of the algebra.

You had another question about the absence of the $\gamma$ matrices. They are there, just written out explicitly.

This post has been migrated from (A51.SE)
answered Mar 24, 2012 by Sidious Lord (160 points) [ no revision ]
Can you give some more explicit expressions about the projections you mentioned or a reference(pedagogic?)? I don't get it when you say that the $\theta_1,\theta_2$ superspace is equally captured by just $\theta_+$ or $\theta_-$. I guess you mean to say that $\theta_+$ and $\bar{\theta}_+$ is the same space as $\theta_1$ and $\theta_2$. So what is the $(2,2)$ superspace. As for extra details - well - much of my immediate notation is from this paper - http://arxiv.org/abs/hep-th/9301042 - and one can see this $\pm$ notation also in this paper - http://arxiv.org/abs/1104.0680.

This post has been migrated from (A51.SE)
@user6818 The first paper you cite deals with two dimensions. So your comment about Chern-Simons definitely doesn't apply. In the second reference I can't see any equation of the kind you asked about. If your question is really about 2D, try this [reference](http://arxiv.org/abs/hep-th/0504147), starting at page 52.

This post has been migrated from (A51.SE)
Thanks for the reference. Let me see if that helps completely understand this $\pm$ notation. About my second reference looks at page 29, 37 and 40 for example - you can see this curious $\pm$ notation - I don't understand what they call as the supersymmetry transformation in that language - like say on page 37.

This post has been migrated from (A51.SE)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights